Conditional-averaging procedure for problems with mode-mode coupling

Abstract

A field is represented by its Fourier modes u(k, t) on the wave-number interval 0~k + k,". We seek to reduce the number of degrees of freedom needed to describe the system by eliminating those modes in the band of wave numbers k, k~k, ", the u+, while retaining their average effect on the remaining modes, the u . Because of mode-mode coupling, this requires, in principle, a conditional average over the u+, in a subensemble in which the u are held (approximately) constant. The conditional average can be related to the full ensemble average by means of the decomposition u+ =v++ 6+, where v+ is independent of u but has the same global mean properties as u+, while h, + represents the effect of mode coupling on the conditional average. Implementation of the formalism for any particular physical system requires an ansatz for the relationship between v+ and u+ such that the conditional average of h, + is small. This procedure is illustrated by its application to the Navier-Stokes equation, where v+ is taken to be the first-order expansion of u+ in a Taylor series about k = k,", thus permitting a renormalization group ca-lculation of the turbulent effective viscosity, as reported previously [Phys. Rev. Lett. 65, 3281 (1990)]. For systems like thisw, ith deterministic time evolution, the conditional average must be based on the weak, or imprecise, condition that u +4 is held constant, where 4 is small (but not zero) and has zero mean under the conditional average, in order to ensure chaotic u+. We show that our formalism, relating conditional and full ensemble averages, is independent of @ to second order in small quantities.PACS number(s): 05.20. -y, 05.45.+b, 47.25.Cg

Two-field theory of incompressible-fluid turbulence

“…1992 The American Physical Society -+vok2 u (k, t)=M & (k) f d j u&(j, t)u~(kj, t), Bt D tt(k}=5 pk ktt~k~ (3) In order to introduce a statistical treatment, we denote the operation of taking an ensemble average by Dirac brackets, thus ( ). We further restrict our attention to incompressible fluids subject to turbulent velocity fields with zero mean, and consider only such fields that are, in addition, homogeneous, isotropic, and stationary in time.…”

“…A full treatment of the formalism has been given elsewhere [3], and we shall concentrate here on the application of the method.…”

“…Taking the conditional average of both sides of Eq. (38}, I it may be shown [3] or it is intuitively obvious that (u+(k, t))0=(U+(k, t))+(b, +(k, t))0 .…”

“…However, it also incorporates a certain amount of new thinking, which not only provides a rational basis for our calculations, but also seems to fit in with other developments in the subject, particularly those having to do with intermittency and ideas of phase correlation. Some details of the calculation have already been given [2], while the formalism underlying our conditional average has recently been explained elsewhere [3].…”

Renormalization group analysis of the magnetohydrodynamic turbulence and dynamo

“…A number of works on renormalization in non-magnetic stirred, stationary turbulence followed, such as e.g. McComb & Watt (1990, 1992), McComb, Roberts & Watt (1992), Rubinstein & Barton (1991, 1992), Smith & Reynolds (1992), Lam (1992) and Eyink (1994) with seminal reviews by Smith & Woodruff (1998) and McComb (2014). A notable contribution came from Kleeorin & Rogachevskii (1994), who calculated the renormalized Lorentz force for non-helical, stationary, homogeneous and isotropic MHD turbulence, in the absence of dynamo action; the effect of chirality in natural turbulence which leads to the -effect and the effect of gradients of the mean fields on the dynamics of fluctuations have been excluded from their analysis.…”